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Full text of "Mathematical And Physical Papers - Iii"

SECTION III.
JVM of the equations in the case of an infinite cylinder oscillating in an unlimited mass of fluid, in a direction perpendicular to >ls axis.
"^ippose a long cylindrical rod suspended at a point in its
ade to oscillate as a pendulum in an unlimited mass of
resistance experienced by any element of the cylinder
etween two parallel planes drawn perpendicular to the
anifestly be very nearly the same as if the element
3 an infinite cylinder oscillating with the same linear
For an element situated very near either extremity of
uhe resistance thus determined would, no doubt, be sensibly
^xVJJ.eous ; but as the diameter of the rod is supposed to be but
small in comparison with its length, it will be easily seen that the
error thus introduced must be extremely small.
Imagine then an infinite cylinder to oscillate in a fluid, in a direction perpendicular to its axis, so that the motion takes place in two dimensions, and let it be required to determine the motion of the fluid. The mode of solution of this problem will require no explanation, being identical in principle with that which has been already adopted in the case of a sphere. In the present instance the problem will be found somewhat easier, up to the formation of the equations analogous to (33) and (34), after which it will become much more difficult.
25. Let a plane drawn perpendicular to the axis of the cylinder be taken for the plane of xy, the origin being situated in the mean position of the axis of the cylinder, and the axis of w being measured in the direction of the cylinder's motion. The general equations (2), (3) become in this case
dp _    fd*u    d*u\      du
~'~~~
(66),
ffv    d?v\      dv dx^d
du    dv
pdt